One pendulum to run them all

TL;DR

This paper derives an analytical solution for the three-dimensional linear pendulum in a rotating frame, unifying Foucault and Bravais pendula, and reveals a new pattern in the Foucault pendulum attractor.

Contribution

It provides the first analytical solution for the 3D linear pendulum including Coriolis and centrifugal effects, unifying different pendulum models.

Findings

Unified treatment of Foucault and Bravais pendula

Analytical expressions in terms of initial conditions

Discovery of a new pattern in the Foucault pendulum attractor

Abstract

The analytical solution of the three--dimensional linear pendulum in a rotating frame of reference is obtained, including Coriolis and centrifugal accelerations, and expressed in terms of initial conditions. This result offers the possibility of treating Foucault and Bravais pendula as trajectories of the very same system of equations, each of them with particular initial conditions. We compare with the common two--dimensional approximations in textbooks. A previously unnoticed pattern in the three--dimensional Foucault pendulum attractor is presented.